Qualitative Modelling and Simulation of Genetic Regulatory Networks in Bacteria Hidde de Jong 1 Delphine Ropers 1 Johannes Geiselmann 1,2 1 INRIA Grenoble-Rhône-Alpes 2 Laboratoire Adaptation Pathogénie des Microorganismes CNRS UMR 5163 Université Joseph Fourier Email: Hidde.de-Jong@inrialpes.fr
� � � IBIS at INRIA Grenoble – Rhône-Alpes IBIS: systems biology group at INRIA and Joseph Fourier University/CNRS Analysis of bacterial regulatory networks Interdisciplinary approach using models and experiments Group composed of computer scientists, biologists, physicists, … 2
Overview 1. Genetic regulatory networks in bacteria 2. Qualitative simulation of genetic regulatory networks using piecewise-linear models 3. Qualitative simulation of carbon starvation response in Escherichia coli : model predictions and validation 4. Conclusions and perspectives 3
Escherichia coli stress responses E. coli is able to adapt to a variety of stresses in its environment Model organism for understanding of decision-making processes in single-cell organisms E. coli is easy to manipulate in the laboratory Model organism for understanding adaptation of pathogenic bacteria to their host Storz and Hengge-Aronis (2000), Bacterial Stress Responses , ASM Press Heat shock Nutritional stress Cold shock Osmotic stress … 2 µm 4
Response of E. coli to carbon starvation Response of E. coli to carbon starvation conditions: transition from exponential phase to stationary phase Changes in morphology, metabolism, gene expression, … log (pop. size) > 4 h time 5
Carbon starvation response network Response of E. coli to carbon source availability is controlled by large and complex genetic regulatory network Ropers et al. (2006), Biosystems , 84(2):124-52 Network senses carbon source availability and global regulators coordinate adaptive response of bacteria 6
Carbon starvation response network Response of E. coli to carbon source availability is controlled by large and complex genetic regulatory network Ropers et al. (2006), Biosystems , 84(2):124-52 No global view of functioning of network available, despite abundant knowledge on network components M athematical modeling and computer analysis and simulation 7
� � � Modeling of genetic regulatory network Well-established theory for modeling of genetic regulatory networks using ordinary differential equation (ODE) models de Jong (2002), J. Comput. Biol. , 9(1): 69-105 Hasty et al. (2001), Nat. Rev. Genet. , 2(4):268-279 Smolen et al. (2000), Bull. Math. Biol. , 62(2):247-292 Szallassi et al. (2006), System Modeling in Cellular Biology , MIT Press Practical problems encountered by modelers: Knowledge on molecular mechanisms rare Quantitative information on kinetic parameters and molecular concentrations absent Large models Even in the case of well-studied E. coli network! 8
� � � Qualitative modeling and simulation Possible strategies to overcome problems Parameter estimation from experimental data Parameter sensitivity analysis Model simplifications Intuition: essential properties of network dynamics robust against reasonable model simplifications Qualitative modeling and simulation of large and complex genetic regulatory networks using simplified models de Jong, Gouzé et al. (2004), Bull. Math. Biol. , 66(2):301-40 Relation with discrete, logical models of gene regulation Thomas and d’Ari (1990), Biological Feedback, CRC Press Kauffman (1993), The Origins of Order, Oxford University Press 9
PL differential equation models Genetic networks modeled by class of differential equations using step functions to describe regulatory interactions . s - ( x , ) x a = κ a s - ( x a , θ a2 ) s - ( x b , θ b ) – γ a x a 1 . x b = κ b s - ( x a , θ a1 ) – γ b x b 0 x θ x : protein concentration θ : threshold concentration κ , γ : rate constants Differential equation models of regulatory networks are piecewise-linear (PL) Glass and Kauffman (1973), J. Theor. Biol. , 39(1):103-29 10
Mathematical analysis of PL models Analysis of local dynamics of PL models Monotonic convergence towards focal state in every region between thresholds max b max b . x a = κ a – γ a x a κ b /γ b . x b = κ b – γ b x b θ b θ b D 1 θ a1 θ a1 κ a /γ a max a max a θ a2 θ a2 0 0 . x a = κ a s - ( x a , θ a2 ) s - ( x b , θ b ) – γ a x a . x b = κ b s - ( x a , θ a1 ) – γ b x b Glass and Kauffman (1973), J. Theor. Biol. , 39(1):103-29 11
Mathematical analysis of PL models Analysis of local dynamics of PL models Monotonic convergence towards focal state in every region between thresholds max b max b . x a = κ a – γ a x a . x b = – γ b x b θ b θ b D 5 θ a1 θ a1 κ a /γ a max a max a θ a2 θ a2 0 0 . x a = κ a s - ( x a , θ a2 ) s - ( x b , θ b ) – γ a x a . x b = κ b s - ( x a , θ a1 ) – γ b x b Glass and Kauffman (1973), J. Theor. Biol. , 39(1):103-29 12
Mathematical analysis of PL models Analysis of local dynamics of PL models max b max b θ b θ b D 3 θ a1 θ a1 max a max a 0 0 θ a2 θ a2 . x a = κ a s - ( x a , θ a2 ) s - ( x b , θ b ) – γ a x a . x b = κ b s - ( x a , θ a1 ) – γ b x b Extension of PL differential equations to differential inclusions using Filippov approach Gouzé and Sari (2002), Dyn. Syst. , 17(4):299-316 13
Mathematical analysis of PL models Analysis of local dynamics of PL models max b max b θ b θ b D 7 θ a1 θ a1 max a max a 0 0 θ a2 θ a2 . x a = κ a s - ( x a , θ a2 ) s - ( x b , θ b ) – γ a x a . x b = κ b s - ( x a , θ a1 ) – γ b x b Extension of PL differential equations to differential inclusions using Filippov approach Gouzé and Sari (2002), Dyn. Syst. , 17(4):299-316 14
Qualitative analysis of PL models State space can be partitioned into regions with unique derivative sign pattern Qualitative abstraction yields state transition graph that provides discrete picture of continuous dynamics D 21 D 24 D 20 max b max b D 25 D 26 D 27 D 19 D 23 D 18 D 24 D 20 D 21 D 17 D 22 D 18 D 19 D 23 D 27 D 25 D 26 D 16 D 17 D 16 D 22 θ b θ b D 13 D 11 D 13 D 14 D 15 D 10 D 12 D 14 D 15 D 10 D 11 D 12 . x a > 0 D 5 : . D 1 D 3 D 5 D 7 D 9 x b < 0 D 1 D 5 D 7 D 9 D 3 D 2 D 4 D 6 D 8 D 2 D 4 D 6 θ a1 θ a1 max a max a θ a2 θ a2 0 0 D 8 . . . x a = 0 x a > 0 x a > 0 . . D 7 : . D 5 : D 1 : x b < 0 x b < 0 de Jong et al. (2004), Bull. Math. Biol. , 66(2):301-40 x b > 0 Batt et al. (2008), Automatica , 44(4):982-89 15
� � Qualitative analysis of PL models State transition graph gives conservative approximation of continuous dynamics Every solution of PL model corresponds to path in state transition graph Converse is not necessarily true! State transition graph is invariant for given inequality constraints on parameters max b max b κ b /γ b 0 < θ a1 < θ a2 < κ a / γ a < max a D 11 D 12 θ b θ b D 11 D 12 0 < θ b < κ b / γ b < max b D 1 D 3 D 1 D 3 θ a1 θ a1 κ a /γ a max a max a θ a2 θ a2 0 0 Batt et al. (2008), Automatica , 44(4):982-89 16
� � Qualitative analysis of PL models State transition graph gives conservative approximation of continuous dynamics Every solution of PL model corresponds to path in state transition graph Converse is not necessarily true! State transition graph is invariant for given inequality constraints on parameters max b max b κ b /γ b 0 < θ a1 < θ a2 < κ a / γ a < max a D 11 D 12 θ b θ b D 11 D 12 0 < θ b < κ b / γ b < max b D 1 D 3 D 1 D 3 θ a1 θ a1 κ a /γ a max a max a θ a2 θ a2 0 0 Batt et al. (2008), Automatica , 44(4):982-89 17
� � Qualitative analysis of PL models State transition graph gives conservative approximation of continuous dynamics Every solution of PL model corresponds to path in state transition graph Converse is not necessarily true! State transition graph is invariant for given inequality constraints on parameters max b max b κ b /γ b D 11 0 < κ a / γ a < θ a1 < θ a2 < max a θ b θ b D 11 D 12 0 < θ b < κ b / γ b < max b D 1 D 3 D 1 κ a /γ a θ a1 θ a1 max a max a θ a2 θ a2 0 0 Batt et al. (2008), Automatica , 44(4):982-89 18
� � Use of state transition graph Analysis of steady states and limit cycles of PL models Attractor states in graph correspond (under certain conditions) to stable steady states of PL model Casey et al. (2006), J. Math Biol. , 52(1):27-56 Attractor cycles in graph correspond (under certain conditions) to stable limit cycles of PL model Glass and Pasternack (1978), J. Math Biol. , 6(2):207-23 Edwards (2000), Physica D , 146(1-4):165-99 D 21 D 24 D 20 max b max b D 25 D 26 D 27 D 19 D 23 D 18 D 17 D 22 D 16 θ b θ b D 13 D 10 D 11 D 12 D 14 D 15 D 1 D 5 D 7 D 9 D 3 D 2 D 4 D 6 θ a1 θ a1 max a max a θ a2 θ a2 0 0 D 8 19
Use of state transition graph Determine reachability of attractors of PL models from given initial states: qualitative simulation Kuipers (1994), Qualitative Reasoning, MIT Press D 21 D 24 D 20 D 25 D 26 D 27 D 19 D 23 D 18 D 17 D 22 D 16 D 13 D 11 D 10 D 12 D 14 D 15 D 1 D 5 D 7 D 9 D 3 D 2 D 4 D 6 D 8 Simulation algorithms based on symbolic computation instead of numerical simulation Use of inequality constraints between parameters 20
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